#183118

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,9,13)(7,11,14,20)(8,16,17,12)(10,18,19,15)(21,22,25,29)(23,31,32,36)(24,30,26,28)(27,34,35,33);(5,7)(6,10,13,19)(8,15)(9,14)(11,16,20,12)(17,18)(21,23,26,34,25,32,24,33)(22,27,30,36,29,35,28,31);(1,3,4,2)(5,7)(6,11)(8,15)(9,14)(10,16)(12,19)(13,20)(17,18);(7,14)(10,19)(11,20)(15,18)(23,32)(27,35)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,e^2=h,g^2=i,b^a=be,e^a=eh,g^a=gh,d^b=dh,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90314);